| L(s) = 1 | + (−1.23 + 3.80i)2-s + (−7.28 − 5.29i)3-s + (−12.9 − 9.40i)4-s + (22.4 − 51.2i)5-s + (29.1 − 21.1i)6-s + 146.·7-s + (51.7 − 37.6i)8-s + (25.0 + 77.0i)9-s + (167. + 148. i)10-s + (−63.0 + 194. i)11-s + (44.4 + 136. i)12-s + (−108. − 335. i)13-s + (−181. + 557. i)14-s + (−434. + 254. i)15-s + (79.1 + 243. i)16-s + (−164. + 119. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.401 − 0.915i)5-s + (0.330 − 0.239i)6-s + 1.13·7-s + (0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s + (0.528 + 0.470i)10-s + (−0.157 + 0.483i)11-s + (0.0892 + 0.274i)12-s + (−0.178 − 0.550i)13-s + (−0.247 + 0.760i)14-s + (−0.498 + 0.291i)15-s + (0.0772 + 0.237i)16-s + (−0.137 + 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.382689508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.382689508\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 3 | \( 1 + (7.28 + 5.29i)T \) |
| 5 | \( 1 + (-22.4 + 51.2i)T \) |
| good | 7 | \( 1 - 146.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (63.0 - 194. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (108. + 335. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (164. - 119. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (158. - 115. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-765. + 2.35e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-808. - 587. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-4.92e3 + 3.57e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-856. - 2.63e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (3.61e3 + 1.11e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (6.64e3 + 4.82e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (8.98e3 + 6.52e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (4.47e3 + 1.37e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.62e4 + 5.00e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-3.84e4 + 2.79e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (6.32e4 + 4.59e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-3.50e3 + 1.07e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-4.92e3 - 3.57e3i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (1.08e4 - 7.91e3i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-231. + 710. i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-6.88e4 - 5.00e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.07639552378332754628731079655, −10.83923448514138501594594459860, −9.763480006951208670521854429609, −8.491035564949432040660460672864, −7.82603432485308877150280145534, −6.47662044221111040432422423954, −5.25989273212071632353571654898, −4.58916182667285497167432909514, −1.89831711601363136676114002838, −0.57090469592328383824956197482,
1.41800963655907533147045631577, 2.86721359521422032101242107532, 4.37374733412836484861640391185, 5.59705383149442079406665013243, 6.99066687346933440615229280558, 8.264374644598638518814213223770, 9.503795729945514439577787660809, 10.46981345114159531273605126648, 11.27479426661504163799315698846, 11.79849791600820336180609753711
